Smooth Approximation of Lipschitz Maps and Their Subgradients

We derive new representations for the generalised Jacobian of a locally Lipschitz map between finite dimensional real Euclidean spaces as the lower limit (i.e., limit inferior) of the classical derivative of the map where it exists. The new representations lead to significantly shorter proofs for the basic properties of the subgradient and the generalised Jacobian including the chain rule. We establish that a sequence of locally Lipschitz maps between finite dimensional Euclidean spaces converges to a given locally Lipschitz map in the L-topology—that is, the weakest refinement of the sup norm topology on the space of locally Lipschitz maps that makes the generalised Jacobian a continuous functional—if and only if the limit superior of the sequence of directional derivatives of the maps in a given vector direction coincides with the generalised directional derivative of the given map in that direction, with the convergence to the limit superior being uniform for all unit vectors. We then prove our main result that the subspace of Lipschitz C ∞ maps between finite dimensional Euclidean spaces is dense in the space of Lipschitz maps equipped with the L-topology, and, for a given Lipschitz map, we explicitly construct a sequence of Lipschitz C ∞ maps converging to it in the L-topology, allowing global smooth approximation of a Lipschitz map and its differential properties. As an application, we obtain a short proof of the extension of Green’s theorem to interval-valued vector fields. For infinite dimensions, we show that the subgradient of a Lipschitz map on a Banach space is upper continuous, and, for a given real-valued Lipschitz map on a separable Banach space, we construct a sequence of Gateaux differentiable functions that converges to the map in the sup norm topology such that the limit superior of the directional derivatives in any direction coincides with the generalised directional derivative of the Lipschitz map in that direction.

An improved proximal method with quasi-distance for nonconvex multiobjective optimization problem

Multiobjective optimization is the optimization with several conflicting objective functions. However, it is generally tough to find an optimal solution that satisfies all objectives from a mathematical frame of reference. The main objective of this article is to present an improved proximal method involving quasi-distance for constrained multiobjective optimization problems under the locally Lipschitz condition of the cost function. An instigation to study the proximal method with quasi distances is due to its widespread applications of the quasi distances in computer theory. To study the convergence result, Fritz John’s necessary optimality condition for weak Pareto solution is used. The suitable conditions to guarantee that the cluster points of the generated sequences are Pareto–Clarke critical points are provided.

Gradient estimates for an orthotropic nonlinear diffusion equation

We consider a quasilinear degenerate parabolic equation driven by the orthotropic p-Laplacian. We prove that local weak solutions are locally Lipschitz continuous in the spatial variable, uniformly in time.

Geometrically Convergent Simulation of the Extrema of Lévy Processes

We develop a novel approximate simulation algorithm for the joint law of the position, the running supremum, and the time of the supremum of a general Lévy process at an arbitrary finite time. We identify the law of the error in simple terms. We prove that the error decays geometrically in Lp (for any [Formula: see text]) as a function of the computational cost, in contrast with the polynomial decay for the approximations available in the literature. We establish a central limit theorem and construct nonasymptotic and asymptotic confidence intervals for the corresponding Monte Carlo estimator. We prove that the multilevel Monte Carlo estimator has optimal computational complexity (i.e., of order [Formula: see text] if the mean squared error is at most [Formula: see text]) for locally Lipschitz and barrier-type functions of the triplet and develop an unbiased version of the estimator. We illustrate the performance of the algorithm with numerical examples.

Optimality conditions for nonsmooth multiobjective bilevel optimization using tangential subdifferentials

In combining the value function approach and tangential subdifferentials, we establish necessary optimality conditions of a nonsmooth multiobjective bilevel programming problem under a suitable constraint qualification. The upper level objectives and constraint functions are neither assumed to be necessarily locally Lipschitz nor convex.

Hausdorff and packing dimensions and measures for nonlinear transversally non-conformal thin solenoids

We extend the results of Hasselblatt and Schmeling [Dimension product structure of hyperbolic sets. Modern Dynamical Systems and Applications. Eds. B. Hasselblatt, M. Brin and Y. Pesin. Cambridge University Press, New York, 2004, pp. 331–345] and of Rams and Simon [Hausdorff and packing measure for solenoids. Ergod. Th. & Dynam. Sys.23 (2003), 273–292] for $C^{1+\varepsilon }$ hyperbolic, (partially) linear solenoids $\Lambda $ over the circle embedded in $\mathbb {R}^3$ non-conformally attracting in the stable discs $W^s$ direction, to nonlinear solenoids. Under the assumptions of transversality and on the Lyapunov exponents for an appropriate Gibbs measure imposing thinness, as well as the assumption that there is an invariant $C^{1+\varepsilon }$ strong stable foliation, we prove that Hausdorff dimension $\operatorname {\mathrm {HD}}(\Lambda \cap W^s)$ is the same quantity $t_0$ for all $W^s$ and else $\mathrm {HD}(\Lambda )=t_0+1$ . We prove also that for the packing measure, $0<\Pi _{t_0}(\Lambda \cap W^s)<\infty $ , but for Hausdorff measure, $\mathrm {HM}_{t_0}(\Lambda \cap W^s)=0$ for all $W^s$ . Also $0<\Pi _{1+t_0}(\Lambda ) <\infty $ and $\mathrm {HM}_{1+t_0}(\Lambda )=0$ . A technical part says that the holonomy along unstable foliation is locally Lipschitz, except for a set of unstable leaves whose intersection with every $W^s$ has measure $\mathrm {HM}_{t_0}$ equal to 0 and even Hausdorff dimension less than $t_0$ . The latter holds due to a large deviations phenomenon.

Limit measures and ergodicity of fractional stochastic reaction–diffusion equations on unbounded domains

This paper deals with invariant measures of fractional stochastic reaction–diffusion equations on unbounded domains with locally Lipschitz continuous drift and diffusion terms. We first prove the existence and regularity of invariant measures, and then show the tightness of the set of all invariant measures of the equation when the noise intensity varies in a bounded interval. We also prove that every limit of invariant measures of the perturbed systems is an invariant measure of the corresponding limiting system. Under further conditions, we establish the ergodicity and the exponentially mixing property of invariant measures.